(* generate a random planar polygon, with specified number of sides *)
randomPlanePolygon[n_] := (
(* sample random plane in n-space, as span of A1, A2 *)
A1 = RandomVariate[NormalDistribution[0,1], n];
A2 = RandomVariate[NormalDistribution[0,1], n];
(* find orthonormal basis B1, B2 of plane *)
{B1, B2} = Orthogonalize[{A1, A2}];
(* square the complex numbers (B1 + i*B2)^2 = C1 + i*C2 *)
C1 = B1^2 - B2^2;
C2 = 2 * B1 * B2;
Pts = Transpose[{C1, C2}];
SumPts = Join[{{0,0}},Accumulate[Pts]];
ListLinePlot[SumPts]
)
randomPlanePolygon[12]
randomPlanePolygon[500]